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In Mathematica the functions like Thread, Inner, Outer etc. are very important and are used frequently.

For the function Thread:


Thread[f[{a, b, c}]]
{f[a], f[b], f[c]}


Thread[f[{a, b, c}, x]]
{f[a, x], f[b, x], f[c, x]}


Thread[f[{a, b, c}, {x, y, z}]]
{f[a, x], f[b, y], f[c, z]}

And I understand the Usage1, Usage2, Usage3 easily as well as I use them masterly.

However I always cannot master the usage of Inner and Outer so that I must refer to the Mathematica Documentation every time when I feel I need using them.

I find that I cannot master them owing to that I cannot understand the results of Inner and Outer clearly. Namely, I always forget what construct they generate when executed.

The typical usage cases of Inner and Outer are shown below:


Inner[f, {a, b}, {x, y}, g]
g[f[a, x], f[b, y]]
Inner[f, {{a, b}, {c, d}}, {x, y}, g]
{g[f[a, x], f[b, y]], g[f[c, x], f[d, y]]}


Outer[f, {a, b}, {x, y, z}]
{{f[a, x], f[a, y], f[a, z]}, {f[b, x], f[b, y], f[b, z]}}


  1. How to master the usage Inner and Outer? Namely, how can I use them without referring to the Mathematica Documentation?

  2. How to understand the result of Out[3],Out[4],Out[5] figuratively? Namely, by using graphics or other way.

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I recommend that you download and work through Leonid Shifrin's Mathematica programming: an advanced introduction. It's free and answers a lot of question you ask. –  m_goldberg Aug 20 '14 at 10:12
@m_goldberg,I read the 1-3 chapters of that book three months ago,but owing to some other things, I don't read the latter chapters.Thanks for your suggestion sincerely.I will continue to read it right now.:-) –  ShutaoTang Aug 20 '14 at 10:23

6 Answers 6

up vote 7 down vote accepted

I think of Outer just like nikie showed.

Inner is a generalization of matrix multiplication. I like the picture from the Wikipedia page.

Matrix Multiplication

To calculate an entry of matrix multiplication, you first pair list entries (a11,b12) and (a12,b22). You "times/multiply" those pairs (a11*b12) and (a12*b22), and then you "plus/add" all the results (a11*b12)+(a12*b22). Note that you "times" before you "plus" in matrix multiplication which helps me remember the order of arguments for Inner.

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Animated Mathematica Functions contains cool animated illustrations of the way a number of built-in functions work. Among them are


enter image description here


enter image description here


enter image description here

See also: carmullion's video

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@kguler...I am learning so much this week...nice –  ubpdqn Aug 20 '14 at 12:52
thank you @ubpdqn... me too :) –  kguler Aug 20 '14 at 13:08

Not sure if that's what you're looking for: This is the image I always have in mind for Outer[f,{a,b,c},{x,y,z}]:

enter image description here

args = {{a, b, c}, {x, y, z}};
TableForm[Outer[f, args[[1]], args[[2]]], TableHeadings -> args]
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I think of Outer like nikie's answer shows. Here's a similar view of Inner. Think of the arguments in columns. Apply f to each row and g to the result.

Mathematica graphics

args = {{a, b, c}, {x, y, z}};
Format[g[e__]] := Column[{g, e},
   Dividers -> {None, {False, True, False}}, Alignment -> Center];
Inner[f, Sequence @@ args, g]
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Might I suggest f@@{a,x} etc.? –  Timothy Wofford Aug 20 '14 at 13:49
Thanks. I wanted a divider, but I hate dealing with tables/grids in Mma. I'd thought about f[a, x], too (i.e., no Format-ting). I was trying to emphasize the columns. –  Michael E2 Aug 20 '14 at 13:52
(i = Inner[List, Range@3, Range@3, List]) // MatrixForm;

enter image description here

(o = Outer[List, Range@3, Range@3]) // MatrixForm

enter image description here

p1 = ListLinePlot[i, Mesh -> All, PlotStyle -> Red, PlotTheme -> "Detailed"];
p2 = ListLinePlot[o, Mesh -> All, PlotStyle -> Blue, PlotTheme -> "Detailed"];

Legended[Show[p2, p1, PlotRange -> All], LineLegend[{Red, Blue}, {"Inner", "Outer"}]]

enter image description here

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like you answer+1 –  ubpdqn Aug 20 '14 at 12:04

My understanding about the execution process of MapThread

I find it is easy to understand MapThread by using the inneral built-in function Transpose

MapThread[f, {{a, b, c}, {x, y, z}, {u, v, w}}]

Apply[f, Transpose[{{a, b, c}, {x, y, z}, {u, v, w}}], {1}]

{f[a, x, u], f[b, y, v], f[c, z, w]}

The comprehension of Inner

Learning from Leonid Shifrin's Mathematica programming: an advanced introduction

Thanks for @m_goldberg's suggesstion

Inner[f,list1,list2,g] == Apply[g,MapThread[f,{list1,list2}]]

Example 1:

(*==>g[f[a, x], f[b, y], f[c, z]]*)


Inner[f,list1,list2,g] == Apply[g,f@@@Transpose[{list1,list2}]]


 Apply[g, f @@@ Transpose[{{a, b, c}, {x, y, z}}]]
 (*==>g[f[a, x], f[b, y], f[c, z]]*)
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